Non-linearity in Davenport-Schinzel Sequences Seth Pettie University of Michigan
Isomorphism and Subsequences Political Isomorphism BUSH is isomorphic to GO BUSH GORE C is isomorphic to A THOMAS is isomorphic to SOUTER THOMAS ,NSA,DOD is not isomorphic to NSF,EPA,NIH CI CIA,N IH Happiness via Subsequences WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? TH TH_WHO I E _ V A BE R TA TARJAN W W O R K S HO P Seth Pettie
Isomorphism and Subsequences Political Isomorphism BUSH is isomorphic to GO BUSH GORE C is isomorphic to A THOMAS is isomorphic to SOUTER THOMAS ,NSA,DOD is not isomorphic to NSF,EPA,NIH CI CIA,N IH Happiness via Subsequences WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? TH TH_WHO I E _ V A BE R TA TARJAN W W O R K S HO P Seth Pettie
Isomorphism and Subsequences Political Isomorphism BUSH is isomorphic to GO BUSH GORE C is isomorphic to A THOMAS is isomorphic to SOUTER THOMAS ,NSA,DOD is not isomorphic to NSF,EPA,NIH CI CIA,N IH Happiness via Subsequences WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? TH TH_WHO I E _ V A BE R TA TARJAN W W O R K S HO P Seth Pettie
Isomorphism and Subsequences Political Isomorphism BUSH is isomorphic to GO BUSH GORE C is isomorphic to A THOMAS is isomorphic to SOUTER THOMAS ,NSA,DOD is not isomorphic to NSF,EPA,NIH CI CIA,N IH Happiness via Subsequences WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? TH TH_WHO I E _ V A BE R TA TARJAN W W O R K S HO P Seth Pettie
Isomorphism and Subsequences Political Isomorphism BUSH is isomorphic to GO BUSH GORE C is isomorphic to A THOMAS is isomorphic to SOUTER THOMAS ,NSA,DOD is not isomorphic to NSF,EPA,NIH CI CIA,N IH Happiness via Subsequences WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? TH TH_WHO I E _ V A BE R TA TARJAN W W O R K S HO P Seth Pettie
Isomorphism and Subsequences Political Isomorphism BUSH is isomorphic to GO BUSH GORE C is isomorphic to A THOMAS is isomorphic to SOUTER THOMAS ,NSA,DOD is not isomorphic to NSF,EPA,NIH CI CIA,N IH Happiness via Subsequences WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? TH_WHO O LD_ R_ VE ? TARJAN FOR PR EZ ? Seth Pettie
Isomorphism and Subsequences Political Isomorphism BUSH is isomorphic to GO BUSH GORE C is isomorphic to A THOMAS is isomorphic to SOUTER THOMAS ,NSA,DOD is not isomorphic to NSF,EPA,NIH CI CIA,N IH Happiness via Subsequences WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? TH_WHO O LD_ R_ VE ? T TARJAN N FO FOR PR EZ ? Seth Pettie
Definitions x ⊂ y : x is isomorphic to a subsequence of y Ex( σ ,n) = max |S| : S ∈ {1,…,n} * σ ⊄ S S is | σ |-regular (technical condition) How fast does Ex( σ ,n) grow as a function of n? Seth Pettie
Original application: lower envelopes (1) Give each object (line segment, quadratic, etc.) a symbol (2) Map the lower envelope to a sequence |S| (3) Show |S| ≤ Ex( σ ,n) for some forbidden subseq. σ this sequence does not contain ababa Seth Pettie
Original motivation: lower envelopes (1) Give each object (line segment, quadratic, etc.) a symbol (2) Map the lower envelope to a sequence |S| (3) Show |S| ≤ Ex( σ ,n) for some forbidden subseq. σ length k+2 standard case: σ = ababab…a “order k Davenport-Schinzel sequence” Seth Pettie
Splay trees and Davenport-Schinzel sequences Amortized analysis: Normally pay for time consuming ops with a reduction in potential Seth Pettie
Splay trees and Davenport-Schinzel sequences New kind of amortized analysis: Label nodes that cannot be paid for by other means Transcribe the labels as a sequence S: |S| ≤ Ex( σ ,n) In [SODA’08] σ = abaabba or abababa Thm. n deque operations take O(n α * (n)) time Seth Pettie
Splay trees and Davenport-Schinzel sequences New kind of amortized analysis: Label nodes that cannot be paid for by other means Transcribe the labels as a sequence S: |S| ≤ Ex( σ ,n) … A much better way to end the proof: … where Ex( σ ,n) = O(n) Seth Pettie
Standard Davenport-Schinzel seqs. α = α (n) α is the inverse-Ackermann function Ex( aba , n) n trivial Ex( abab , n) 2n-1 Seth Pettie
Standard Davenport-Schinzel seqs. α = α (n) α is the inverse-Ackermann function Ex( aba , n) n trivial Ex( abab , n) 2n-1 Ex( ababa , n) Θ (n α ) Hart-Sharir Seth Pettie
Standard Davenport-Schinzel seqs. α = α (n) α is the inverse-Ackermann function Ex( aba , n) n trivial Ex( abab , n) 2n-1 Ex( ababa , n) Θ (n α ) Hart-Sharir Ex( ababab , n) Θ (n2 α ) Agarwal-Sharir-Shor Seth Pettie
Standard Davenport-Schinzel seqs. α = α (n) α is the inverse-Ackermann function Ex( aba , n) n trivial Ex( abab , n) 2n-1 Ex( ababa , n) Θ (n α ) Hart-Sharir Ex( ababab , n) Θ (n2 α ) Agarwal-Sharir-Shor Ex( abababa , n) n exp(O( α log α )) Ex( abababab , n) n exp( Θ ( α 2 )) Ex( ababababa , n) n exp(O( α 2 log α )) Ex( ababababab , n) n exp( Θ ( α 3 )) Seth Pettie
Standard Davenport-Schinzel seqs. α = α (n) α is the inverse-Ackermann function Ex( aba , n) n trivial Ex( abab , n) 2n-1 Ex( ababa , n) Θ (n α ) Hart-Sharir Ex( ababab , n) Θ (n2 α ) Agarwal-Sharir-Shor Ex( abababa , n) n exp(O( α log α )) Ex( abababab , n) n exp( Θ ( α 2 )) Ex( ababababa , n) n exp(O( α 2 log α )) Ex( ababababab , n) n exp( Θ ( α 3 )) Ex( σ , n) n exp(O( α | σ | )) Klazar Seth Pettie
Two-Letter Forbidden Subsequences [Adamec-Klazar-Valtr] Ex(abbaab,n) = O(n) The Two-Letter Theorem : For any σ ∈ {a,b} * Ex( σ ,n) = ω (n) if and only if ababa ⊂ σ (i.e., there is only one “cause” of superlinearity over two symbols) Seth Pettie
The Three-Letter Theorem [Klazar-Valtr] For σ ∈ {a,b,c} * Ex( σ ,n) = O(n) unless… ababa ⊂ σ or non-linear abcacbc ⊂ σ or status still open abcbcac ⊂ σ or their reversals Seth Pettie
Recipe for linear forbidden sequences [Klazar-Valtr] (1) Ex(a i ,n) = O(n) (2) If Ex( u aa w ,n) = O(n) and Ex( v ,n) = O(n) Ex( u a v a w ,n) = O(n) (3) If Ex( u aa w a) = O(n) Ex( u ab i a w ab i ) = O(n) For Example: Ex( aabbaabcdddcefgfefgcccbbccdd ) = O(n) Seth Pettie
Recipe for linear forbidden sequences [Klazar-Valtr] (1) Ex(a i ,n) = O(n) (2) If Ex( uw ,n) = O(n) and Ex( v ,n) = O(n) uw and v have Ex( uvw ,n) = O(n) disjoint alphabets (3) If Ex( u aa w a) = O(n) Ex( u ab i a w ab i ) = O(n) For Example: Ex( aabbaabcdddcefgfefgcccbbccdd ) = O(n) Seth Pettie
Recipe for linear forbidden sequences [Klazar-Valtr] (1) Ex(a i ,n) = O(n) (2) If Ex( uw ,n) = O(n) and Ex( v ,n) = O(n) Ex( uvw ,n) = O(n) uw and v have disjoint alphabets (3) If Ex( u a w a,n) = O(n) Ex( u ab i w ab i ) = O(n) Seth Pettie
Recipe for linear forbidden sequences [Klazar-Valtr] (1) Ex(a i ,n) = O(n) (2) If Ex( uw ,n) = O(n) and Ex( v ,n) = O(n) Ex( uvw ,n) = O(n) uw and v have disjoint alphabets (3) If Ex( u a w a,n) = O(n) Ex( u ab i w ab i ) = O(n) aaaa Seth Pettie
Recipe for linear forbidden sequences [Klazar-Valtr] (1) Ex(a i ,n) = O(n) (2) If Ex( uw ,n) = O(n) and Ex( v ,n) = O(n) Ex( uvw ,n) = O(n) uw and v have disjoint alphabets (3) If Ex( u a w a,n) = O(n) Ex( u ab i w ab i ) = O(n) aa bb aa bbb Seth Pettie
Recipe for linear forbidden sequences [Klazar-Valtr] (1) Ex(a i ,n) = O(n) (2) If Ex( uw ,n) = O(n) and Ex( v ,n) = O(n) Ex( uvw ,n) = O(n) uw and v have disjoint alphabets (3) If Ex( u a w a,n) = O(n) Ex( u ab i w ab i ) = O(n) aabbaab ccccc bb cc Seth Pettie
Recipe for linear forbidden sequences [Klazar-Valtr] (1) Ex(a i ,n) = O(n) (2) If Ex( uw ,n) = O(n) and Ex( v ,n) = O(n) Ex( uvw ,n) = O(n) uw and v have disjoint alphabets (3) If Ex( u a w a,n) = O(n) Ex( u ab i w ab i ) = O(n) aabbaabc ddd ccccbbcc dd Seth Pettie
Recipe for linear forbidden sequences [Klazar-Valtr] (1) Ex(a i ,n) = O(n) (2) If Ex( uw ,n) = O(n) and Ex( v ,n) = O(n) Ex( uvw ,n) = O(n) uw and v have disjoint alphabets (3) If Ex( u a w a,n) = O(n) Ex( u ab i w ab i ) = O(n) aabbaabcdddccccbbccdd ee Seth Pettie
Recipe for linear forbidden sequences [Klazar-Valtr] (1) Ex(a i ,n) = O(n) (2) If Ex( uw ,n) = O(n) and Ex( v ,n) = O(n) Ex( uvw ,n) = O(n) uw and v have disjoint alphabets (3) If Ex( u a w a,n) = O(n) Ex( u ab i w ab i ) = O(n) aabbaabcdddccccbbccdd e ff e f Seth Pettie
Recipe for linear forbidden sequences [Klazar-Valtr] (1) Ex(a i ,n) = O(n) (2) If Ex( uw ,n) = O(n) and Ex( v ,n) = O(n) Ex( uvw ,n) = O(n) uw and v have disjoint alphabets (3) If Ex( u a w a,n) = O(n) Ex( u ab i w ab i ) = O(n) aabbaabcdddccccbbccdd ef g fef g Seth Pettie
Recipe for linear forbidden sequences [Klazar-Valtr] (1) Ex(a i ,n) = O(n) (2) If Ex( uw ,n) = O(n) and Ex( v ,n) = O(n) Ex( uvw ,n) = O(n) uw and v have disjoint alphabets (3) If Ex( u a w a,n) = O(n) Ex( u ab i w ab i ) = O(n) aabbaabcdddc efgfefg cccbbccdd efgfefg Seth Pettie
More than one cause of non-linearity [Klazar] σ is a sequence without repetitions (x,y) is in G( σ ) iff xyyx ⊂ σ or yxyx ⊂ σ If G( σ ) is strongly connected then Ex( σ ,n) = Ω (n α (n)) Seth Pettie
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